# Christiano-Fitzgerald ﬁlter

### The Band Pass Filter

Optimal finite‐sample approximations for the band pass filter

When I did my master thesis at the University of Cologne on the topic of monetary business cycle theory in 2003 I became interested in the work of Lawrence Christiano. The article regarding the band-pass filter – I did not include it in my thesis – was particularly interesting. After graduating I decided to implement the band-pass filter in an Excel Add-In and I redesigned it in 2019.

This add-in provides some band-pass filters to isolate the cyclical component of a time series by specifying a range for its duration. It is an implementation of the filters presented in the article “The Band Pass Filter” by Lawrence J. Christiano and Terry J. Fitzgerald. They developed optimal finite-sample approximation for the band pass filter, which includes one filter that can be used in real time. For filtering a time series, the user must first choose the range ($p_l$ and $p_u$) of durations to pass through.

In example:

• Quarterly data: $p_l$=6, $p_u$=32 returns component with periods between 1.5 and 8 yrs.
• Monthly data: $p_l$=2, $p_u$=24 returns component with all periods less than 2 yrs.

Some of the filters provide you with additional options for handling trending data and I(1) processes.

In the add-in following filters are available:

1. Full sample asymmetric Christiano-Fitzgerald filter
2. Symmetric Christiano-Fitzgerald filter
3. Fixed length symmetric Christiano-Fitzgerald filter
4. Fixed length symmetric Baxter-King filter
5. Trigonometric regression filter

For further details on the theoretical background, please refer to the Article “The Band Pass Filter” by Lawrence J. Christiano and Terry J. Fitzgerald.

The following table shows the valid parameter combinations:

Filter$p_l$$p_u$$I(1)$trendfixed lengthMA coefficientsmisc
Asymmetric CF>=2>$p_l$OptionalOptional./.Optional./.
Symmetric CF>=2>$p_l$OptionalOptional./.Optional./.
Fixed Length sym-metric CF>=2>$p_l$OptionalOptionalRequiredOptional./.
Fixed Length sym-metric BK>=2>$p_l$./.OptionalRequired./../.
Trigonometric>=2>$p_l$./.Optional./../.available for even T